 Bounded-radius propagation of edge defects

In an embedded arrangement graph, an edge whose fixed-radius neighbourhood meets a bad edge can be charged to the first such edge along a shortest path. All intervening edges are good dual-arrangement edges, so their endpoints have bounded degree. The number of possible reversed paths is bounded in terms of the fixed radius alone. Therefore thickening a defect set by a fixed radius preserves an $O(t_2)$ edge count.